Any frequentist confidence interval based on a normal approximation may give puzzling results when there are $0$ successes in $n$ trials.
It might be better to get a Bayesian probability interval estimate (sometimes called 'credible interval') based on a suitable beta prior distribution. If you have little or no prior information, you might use $\mathsf{Unif}(0,1) \equiv \mathsf{Beta}(1,1)$ as the prior. An alternative would be to use the Jeffreys prior $\mathsf{Beta}(.5, .5).$
If getting no successes does not come as a surprise, then perhaps you want to use a prior distribution on the success probability that puts more probability toward the lower end of $(0,1).$
Suppose you use a uniform prior and get $x = 0$ successes in $n = 20$ trials. Then the posterior distribution is $\mathsf{Beta}(1+x, 1+n-x) - \mathsf{Beta}(1,21).$ Then a one-sided 95% Bayesian interval estimate $(0, 0.0325)$ can be found in R as shown below. This interval suggests that the small success probability is likely below $0.0325.$
qbeta(.5, 1, 21)
[1] 0.03246822
If you really want a two-sided interval estimate for the binomial success probability, then you could use $(0.0012, 0.1611).$ [The Jeffries 95% two-sided interval is $(0.000024, 0.11664).]$
qbeta(c(.025, .975), 1, 21) # uniform prior
[1] 0.001204883 0.161097615
qbeta(c(.025, .975), .5, 20.5) # Jeffries prior
[1] 2.424648e-05 1.166390e-01
This interval makes it clear that you are not suggesting that the events in question are completely impossible. For example, if the task is to estimate the proportion of cattle in Montana that have a particular disease, you are not saying that negative results from 20 randomly chosen Montana animals should be taken as an indication that cattle throughout Montana are totally free of the disease.
There are important philosophical differences between Bayesian and frequentist statisticians, and these differences extend to how interval estimates are interpreted by both groups of statisticians. However, frequentist statisticians are often happy to use Bayesian computational methods for getting an interval estimate, and then calling the result a 'confidence interval' to be given a frequentist interpretation.
Note: For an extensive list of types of frequentist CIs for the binomial success probability see Wikipedia, where assumptions are discussed. Notice that the Jeffreys interval, interpreted as frequentist, is among the first types of CIs discussed there.